Download VECTOR ANALYSIS In this course, vectors like electric and

VECTOR ANALYSIS
In this course, vectors like electric and magnetic fields will be discussed
both in free space and in material media.
In order to do this, some numerical and mathematical tools; like addition,
multiplication, differentiation and integration of scalars and vectors will be
used.
SCALARS:
A scalar is s quantity that has only magnitude. i.e. mass, temperature,
electric potential, charge density.
VECTORS:
A vector is a quantity that has both magnitude and direction. i.e. force,
velocity, field.
A field is a function that specifies a particular quantity everywhere in a
region.
In electromagnetics, common fields that will be used are the electric field
intensity E , electric flux density D (displacement vector), magnetic field
intensity vector H and the magnetic flux density vector B .
VECTOR ALGEBRA
Vector algebra governs the laws of addition, subtraction and multiplication
of vectors in any given coordinate system.
Vector Addition: Consider two vectors
A
and B .
Their sum is: C = A + B
We can add these vectors graphically to form vector C .
Figs.1-1. a and b shows Parallelogram and Head to Tail Rules for adding vectors.
Properties of addition:
i)
Addition is commutative, i.e.
A1 + A2 = A2 + A1
ii)
Addition is associative i.e.
(A + A )+ A
1
iii)
iv)
2
3
= A1 + ( A2 + A3 ) = A1 + A2 + A3
On the side two sides of vector equality we can add the same
vector. i.e.
If A = B , then
A+C = B +C
The magnitude of the sum of N vectors is less than or equal to the
sum of their magnitudes. i.e.
A1 + A2 + ... + AN ≤ A1 + A2 + ... + AN
If the N vectors are all in the same direction and have the same senses, then
the equality holds.
Multiplication of a Vector by a Scalar
Consider a vector A and a scalar λ . We define the multiplication of a vector
A by a scalar λ as the vector B such that,
a) if A or λ is zero (or both), then B = 0 ,
b) if A ≠ 0 and λ ≠ 0 , then
i)
B has the same direction of
ii)
If λ > 0 , B and
iii)
If λ < 0 ,
iv)
Magnitude of B is, B =
A
A have the same sense
B and A have opposite senses.
λ A
We denote the multiplication as:
B = λA
Consider two non-zero vectors. If they have the same direction, then they are
said to be parallel.
λ A are parallel to A . Conversely, every
A has the form of λ A .
Then, all vectors of the form
vector which is parallel to
VECTOR SUBTRUCTION
Vector subtraction is D = A − B .
Figs.1-2. a and b shows Parallelogram and Head to Tail Rules for subtracting vectors.
UNIT VECTOR:
A unit vector is a vector of unity magnitude. A unit vector in the same
direction (with the same sense) of a vector
A is:
A
A
If we denote this unit vector as
aˆ A =
We can write
A
aˆ A , then,
A
A
by multiplying each side by A ,
A = aˆ A A
Cartesian Components of a Vector
Consider the Cartesian coordinate system.
Fig. 1-3
Denote the unit vectors along the
x , y and z
Then, any vector along the x − axis is:
Ax = Ax aˆ x
Similarly Ay = Ay aˆ y and
Az = Az aˆ z .
Then any vector A can be written as:
A = Ax aˆ x + Ay aˆ y + Az aˆ z
The magnitude of A is:
A = Ax 2 + Ay 2 + Az 2
axes as
aˆ x
,
aˆ y and aˆ z .
VECTOR MULTIPLICATION
I)
SCALAR (DOT) MULTIPLICATION
Let A and B be two non-zero vectors. Let the angle between A and B be
θ AB . By the scalar product of these two vectors, we mean the following
scalar:
A B cos θ AB
The scalar product of A and B is denoted as A.B . Then,
A.B = A B cos θ AB
A.B is a signed scalar.
A ≠ 0 and B ≠ 0 , if 0 ≤ θ AB <
Let
π
π
A.B > 0
2 then
if 2 < θ AB ≤ π then A.B < 0
π
θ
=
if AB 2 , cos θ AB = 0 and A.B = 0
Scalar Product of Two Vectors in Terms of Their Cartesian
Components
Let A = Ax aˆ x + Ay aˆ y + Az aˆ z and
B = Bx aˆ x + B y aˆ y + Bz aˆ z
(
)(
Then, A.B = Ax aˆ x + Ay aˆ y + Az aˆ z . Bx aˆ x + B y aˆ y + Bz aˆ z
A.B = A x Bx + A y By + A z Bz
Since,
aˆ x .aˆ x = aˆ y .aˆ y = aˆ z .aˆ z = 1
)
aˆ x .aˆ y = aˆ x .aˆ z = aˆ y .aˆ z = 0
II)
VECTOR (CROSS) PRODUCT
Consider two arbitrary vectors A and B . We define the vector or cross
product of A and B as a new vector C with the following properties:
If either A and B (or both) is zero then C = 0 . Now consider,
i)
A≠0,B ≠0
ii)
The direction of C is perpendicular to the plane which is formed by
iii)
A and B , hence C is perpendicular to both A and B .
Let the angle between A and B be θ . The magnitude of C is:
C = A B sin θ (since 0 ≤ θ ≤ π , sin θ ≥ 0 ), in other words, C
is equal to the area of the parallelogram formed by A and B :
Fig. 1-4 The cross product of A and B is a vector with magnitude equal
to the area of the parallelogram and direction as indicated.
iv)
The sense of C is determined by the right hand rule.
We denote the vector product of two vectors as: AXB = A B sin θ aˆn where
aˆ n is the unit vector perpendicular to both A and B .
AXB = 0 if A and B are parallel.
v)
vi)
AXB = − BXA
vii)
AXB = BXA
Fig. 1-5 Direction of AXB and aˆ n using a) right-hand rule b) righthanded screw rule.
Representation of AXB in Terms of the Cartesian Components
From the definition of the cross-product, we can write:
aˆ x X aˆ y = aˆ z
aˆ y X aˆ z = aˆ x
aˆ z X aˆ x = aˆ y
AXB = ( Ax aˆ x + Ay aˆ y + Az aˆ z ) X ( Bx aˆ x + B y aˆ y + Bz aˆ z )
We have a compact form for the AXB in terms of a determinant:
aˆ x
AXB = Ax
Bx
aˆ y
Ay
By
aˆ z
Az
Bz
III)
Scalar Triple Product
Given three vectors A , B and C , we define the scalar triple product as:
A. ( BXC ) = B . ( CXA ) = C . ( AXB )
A.( BXC )
is the volume of the parallelepiped having A , B and C as edges
and is easily obtained by finding the determined:
Ax
A.( BXC ) = Bx
Cx
A.( BXC ) = 0 if
A , B and C are coplanar.
Ay
Az
By
Cy
Bz
Cz